Requirement came first, making one annoyed (7)
I believe the answer is:
needled
'one annoyed' is the definition.
(needle can mean to annoy or irritate)
'requirement came first' is the wordplay.
'requirement' becomes 'need' (synonyms).
'came first' becomes 'led' (I've seen this before).
'need'+'led'='NEEDLED'
'making' is the link.
(Other definitions for needled that I've seen before include "Piqued" , "got angry" , "Provoked with sewing implement" , "Annoyed - provoked" , "bugged" .)